Definition

Note that if P is true, then Q must also be true for the implication to hold (be true). However, if P is false, Q may or may not be true and the implication still holds.

To illustrate the latter fact, consider a teacher who tells her class that any student who gets 100% on the final exam will pass the class. In other words:

P: A student gets 100% on the final.

Q: That student passes the class.

P → Q: If a student gets 100% on the final, then that student passes the class.

Now consider the case in which two students do poorly on the final exam; one of them did well enough on the other exams to pass the course, but the other did not. Did the teacher lie?

No. She said nothing about students who do not get 100% on the final (i.e., the case where P is false). Unless there is a student who both got 100% on the final and did not pass the course, the teacher told the truth.

For this reason, "P → Q" can be restated as "¬(P ∧ ¬ Q)" or "not (P and not Q)", or "it is not the case that P is true and Q is false".

If the last statement were not equivalent to the original, then I might be able to not study and still pass the class — but this contradicts my original assertion that it was only by studying hard that I would pass.

Using an "unless" statement to say the same thing:

I won't pass the class unless I study hard.

This means I'll want to study hard to give myself a chance to pass, but it doesn't guarantee that I will pass; on the other hand, not studying will guarantee that I don't pass.

Symbolically, using

S = I (do/will) study hard.

P = I (do/will) pass the class.

all of these statements can be written as

¬ S → ¬ P: If I don't study hard, then I won't pass the class.

or, equivalently

P → S: If I do pass the class then I will study hard.

Rephrasing the last statement to make more sense grammatically:

If I end up passing the class, then I must have studied hard.

Transformations of conditionals

Given a conditional statement "if P then Q", there are many ways of transforming the statement to other conditionals that may or may not be logically equivalent.